By Professor Oscar Gonzalez, Professor Andrew M. Stuart
A concise account of vintage theories of fluids and solids, for graduate and complicated undergraduate classes in continuum mechanics.
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Additional resources for A First Course in Continuum Mechanics
Let G : V 2 → V 2 be an isotropic function which maps symmetric tensors to symmetric tensors, that is, G(A) ∈ V 2 is symmetric if A ∈ V 2 is symmetric. Then there are functions α0 , α1 , α2 : IR3 → IR such that G(A) = α0 (IA )I + α1 (IA )A + α2 (IA )A2 for every symmetric A ∈ V 2 . 10, there are functions β0 , β1 , β2 : IR3 → IR such that G(A) = β0 (IA )I + β1 (IA )A + β2 (IA )A−1 for every symmetric A ∈ V 2 which is invertible. The above result may be strengthened when, in addition to the properties mentioned above, G is linear and satisﬁes G(W ) = O for every skew-symmetric W .
As we will see, a fourthorder tensor will be described by eighty-one components in any Cartesian coordinate frame for IE 3 . 1 Deﬁnition By a fourth-order tensor C on the vector space V we mean a mapping C : V 2 → V 2 which is linear in the sense that: (1) C(S + T ) = CS + CT for all S, T ∈ V 2 , (2) C(αT ) = αCT for all α ∈ IR and T ∈ V 2 . We denote the set of all fourth-order tensors on V by the symbol V 4 . We deﬁne a fourth-order zero tensor O with the property OT = O for all T ∈ V 2 , and we deﬁne a fourth-order identity tensor I with the property IT = T for all T ∈ V 2 .
For other examples see the discussion of the trace, determinant and principal invariants of second-order tensors. 5 Epsilon-Delta Identities By virtue of their deﬁnitions in terms of a right-handed orthonormal frame, the permutation symbol and the Kronecker delta satisfy the following identities. 1 Epsilon-Delta Identities. Let symbol and δij the Kronecker delta. Then pq s n r s = δpn δq r − δpr δq n Proof See Exercise 18. and ij k pq s r q s be the permutation = 2δpr . 3 Second-Order Tensors 11 As an application of the above result we establish an identity involving the triple vector product a × (b × c) = (a · c)b − (a · b)c, ∀a, b, c ∈ V.